Based on the family the graph below belongs to, which equation could represent the graph?
1 answer:
Considering the asymptotes of the function, the equation that represents the graph is:
- The vertical asymptotes are the values of x which are outside the domain, which in a fraction are the zeroes of the denominator.
- The horizontal asymptote is the value of f(x) as x goes to infinity, as long as this value is different of infinity.
This function, from the graph, has a vertical asymptote at x = -2, hence the denominator is given by:
x + 2, as x + 2 = 0 -> x = -2.
The horizontal asymptote is of y = 3, hence:
Which means that the function is described by the following rule:
More can be learned about asymptotes at brainly.com/question/16948935
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The slope of tangent line at point(4, 2pi) is undefined.
For given question,
We have been given a polar equation r = 2 + 2cos(θ)
We need find dy/dx as well as the slope of tangent line at point(4, 2π).
We know that, for polar equation we use,
x = r cos(θ) and y = r sin(θ)
plug the given value of r into these equations we get:
⇒ x = r cos(θ)
⇒ x = (2 + 2cos(θ) ) × cos(θ)
⇒ x = 2(cos(θ) + cos²(θ))
⇒ x = 2cos(θ) + 2cos²(θ)
Similarly,
⇒ y = r sin(θ)
⇒ y = (2 + 2cos(θ) ) × sin(θ)
⇒ y = 2(sin(θ) + sin(θ)cos(θ))
⇒ y = 2sin(θ) + 2sin(θ)cos(θ)
Now we find derivative of x and y with respect to theta.
.............(1)
Similarly,
..............(2)
Now we find dy/dx
⇒ dy/dx = (dy/dθ) / (dx/dθ)
From (1) and (2),
We know that The slope of tangent line is given by dy/dx.
So, the slope is:
Now we need to find the slope of tangent line at point(4, 2pi)
Substitute θ = 2π in above slope formula.
⇒ m = ∞
The slope of tangent line at point(4, 2pi) is not defined.
This means, the tangent line must be parallel to Y-axis.
Therefore, the slope of tangent line at point(4, 2pi) is undefined.
Learn more about the slope here:
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